Decomposing data sets into skewness modes

We derive the nonlinear equations satisfied by the coefficients of linear combinations that maximize their skewness when their variance is constrained to take a specific value. In order to numerically solve these nonlinear equations we develop a gradient-type flow that preserves the constraint. In combination with the Karhunen–Loève decomposition this leads to a set of orthogonal modes with maximal skewness. For illustration purposes we apply these techniques to atmospheric data; in this case the maximal-skewness modes correspond to strongly localized atmospheric flows. We have also checked that the results are statistically significant in spite of the finite length of the data. We show how these ideas can be extended, for example to maximal-flatness modes.